Nonlinear dynamical systems: Time reversibility {\it versus} sensitivity to the initial conditions

TL;DR

This paper investigates the relationship between time reversibility and sensitivity to initial conditions in a nonlinear dynamical system, revealing conditions under which past information can be recovered, with implications across various scientific fields.

Contribution

It demonstrates how time reversibility correlates with the Lyapunov exponent in the logistic map, highlighting scenarios where past states can be reconstructed despite chaos.

Findings

Time reversal enables past state recovery at non-positive Lyapunov exponents.

Weak chaos at the Feigenbaum point allows for near-perfect time reversal.

Strong chaos with positive Lyapunov exponents leads to loss of past information.

Abstract

Time reversal of vast classes of phenomena has direct implications with predictability, causality and the second principle of thermodynamics. We analyze in detail time reversibility of a paradigmatic dissipative nonlinear dynamical system, namely the logistic map $x_{t+1}=1-ax_t^2$. A close relation is revealed between time reversibility and the sensitivity to the initial conditions. Indeed, depending on the initial condition and the size of the time series, time reversal can enable the recovery, within a small error bar, of past information when the Lyapunov exponent is non-positive, notably at the Feigenbaum point (edge of chaos), where weak chaos is known to exist. Past information is gradually lost for increasingly large Lyapunov exponent (strong chaos), notably at $a=2$ where it attains a large value. These facts open the door to diverse novel applications in physicochemical, astronomical, medical, financial, and other time series.